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SOLUTIONS OF THE VON KÁRMÁN PLATE EQUATIONSBY A GALERKIN METHOD, WITHOUT INVERTING

THE TANGENT STIFFNESS MATRIX

Honghua Dai, Xiaokui Yue and Satya N. Atluri

Volume 9, No. 2 March 2014

msp

JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURESVol. 9, No. 2, 2014

dx.doi.org/10.2140/jomms.2014.9.195 msp

SOLUTIONS OF THE VON KÁRMÁN PLATE EQUATIONSBY A GALERKIN METHOD, WITHOUT INVERTING

THE TANGENT STIFFNESS MATRIX

HONGHUA DAI, XIAOKUI YUE AND SATYA N. ATLURI

Large deflections of a simply supported von Kármán plate with imperfect initial deflections, under acombination of in-plane loads and lateral pressure, are analyzed by a semianalytical global Galerkinmethod. While many may argue that the dominance of the finite element method in the marketplacemay make any other attempts to solve nonlinear plate problems to be redundant and obsolete, semi- andprecise analytical methods, when possible, simply serve as benchmark solutions if nothing else. Also,since parametric variations are simpler to access through such analytical methods, they are more usefulin studying the physics of the phenomena. In the present method, the Galerkin scheme is first appliedto transform the governing nonlinear partial differential equations of the von Kármán plate into a systemof general nonlinear algebraic equations (NAEs) in an explicit form. The Jacobian matrix, the tangentstiffness matrix of the system of NAEs, is explicitly derived, which speeds up the Newton–Raphsoniterative method if it is used. The present global Galerkin method is compared with the incrementalGalerkin method, the perturbation method, the finite element method and the finite difference method insolving the von Kármán plate equations to compare their relative accuracies and efficiencies. Bucklingbehavior and jump phenomenon of the plate are detected and analyzed. Besides the classical Newton–Raphson method, an entirely novel series of scalar homotopy methods, which do not need to invert theJacobian matrix (the tangent stiffness matrix), even in an elastostatic problem, and which are insensitiveto the guesses of the initial solution, are introduced. Furthermore, we provide a comprehensive review ofthe newly developed scalar homotopy methods, and incorporate them into a uniform framework, whichrenders a clear and concise understanding of the scalar homotopy methods. In addition, the performanceof various scalar homotopy methods is evaluated through solving the Galerkin-resulting NAEs. Thepresent scalar homotopy methods are advantageous when the system of NAEs is very large in size, whenthe inversion of the Jacobian may be avoided altogether, when the Jacobian is nearly singular, and thesensitivity to the initially guessed solution as in the Newton–Raphson method needs to be avoided, andwhen the system of NAEs is either over- or under-determined.

1. Introduction

Analysis of large deflections of square and rectangular plates is one of the most studied engineeringproblems in the structural community, with many engineering applications including in aircraft structures,shipbuilding, bridges, and spaceships. The thin plates used in aircraft construction are subjected to lateralloads from the pressurized cabin or from the lift on the wings, and to edge loading due to bending of thefuselage and wings. The skin plates of a ship bottom are subjected to a significant water pressure, and to

Keywords: von Kármán plate equations, initial imperfection, global Galerkin method, nonlinear algebraic equations, scalarhomotopy methods, buckling behavior.

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196 HONGHUA DAI, XIAOKUI YUE AND SATYA N. ATLURI

the edge loading owing to bending of the hull. Plate bending problems are also applicable to spaceships,where the outer plates may undergo lateral pressures and in-plane loadings.

The classical Kirchhoff theory for linear plate bending is accurate only for small deflection problems(w ≤ 0.2 thickness) ignoring the middle surface strains and the corresponding in-plane stresses. Asthe external force increases, the lateral deflection may be relatively large (w ≥ 0.3 thickness). In thisscenario, the stretching of the middle surface of the plate should be considered and correspondingly themembrane forces arising from this stretching play a role in carrying lateral loads. The extension to largedeformations was first provided by von Kármán in a seminal work [1910], wherein the nonlinear termsare retained in the kinematic relationships to account for a significantly large deflection of the plate (wis comparable with plate thickness or larger but remains small with respect to other dimensions of theplate). This leads to a pair of coupled nonlinear fourth-order equations for the transverse displacement,and the stress function for the in-plane stress resultants. Inasmuch as the nonlinear terms are included inthe coupled PDEs, closed form solutions for the nonlinear problem do not exist.

The first attempt to solve the von Kármán plate equations by a semianalytical method is attributed toWay [1939], who analyzed a geometrically nonlinear clamped rectangular plate via an energy methodto obtain approximate solutions. Levy [1942a; 1942b] applied a Fourier series method to solve a simplysupported rectangular plate under combined edge compression and lateral loading. Then Levy [1944]and Woolley et al. [1946] analyzed long rectangular plates with simply supported edges and clampededges respectively by a similar approach. Okada, Oshima and Fukumoto [Okada et al. 1979] appliedthe Rayleigh–Ritz method to a simply supported long rectangular plate (length/width = 3 and 4) underhydrostatic pressure, and discussed various buckling behaviors. Ueda, Rashed and Paik [Ueda et al. 1987]proposed an incremental Galerkin method by solving stepwise the linearized form of the von Kármánplate equations of the simply supported rectangular plate. The incremental Galerkin method was thenapplied to solve stiffened ship plates [Paik et al. 2001]. Large deflection of a simply supported plate wasalso analyzed in [Shen 1989] using a perturbation method and in [Bert et al. 1989] using a differentialquadrature method. These semianalytical methods have respective drawbacks in that they may be toocomplex mathematically, or require large amounts of computational efforts, or have a slow convergencerate of the solution. Readers are advised to see [Chia 1980] for a comprehensive review.

With the development of modern digital computers, numerical methods based on domain discretizationtook over the difficult task. Brown and Harvey [1969] used the finite difference method to carry outthe analysis of large deflections of rectangular plates subjected to a combination of lateral pressureand edge loading. Some of the earlier finite element implementations for large deformations wereconducted in [Brebbia and Connor 1969] and [Bergan and Clough 1973], and involved conventionaldisplacement based elements with the strain energy expressed in terms of the three displacement com-ponents. The stress based finite element method was proposed in [Punch and Atluri 1986]. A boundaryelement approach was proposed to investigate static, dynamic and buckling behavior of thin flat plates in[O’Donoghue and Atluri 1987]. Although these numerical methods can be employed to solve the von Kár-mán plate equations accurately and are more flexible than the semianalytical methods with respect tovarious boundary conditions and geometries, these domain discretization methods would require severalorders of magnitude more degrees of freedom than semianalytical approaches. Therefore, the computa-tional burden is comparatively heavy. Also, semianalytical methods will provide the needed benchmarksolutions with minimal computational cost, to judge the accuracies of the many fully numerical methods

SOLUTIONS OF THE VON KÁRMÁN PLATE EQUATIONS BY A GALERKIN METHOD 197

using spatial discretization based on simple polynomials locally.In this paper a simply supported rectangular plate with initial imperfections, under a combination of

in-plane and out of plane loads, is analyzed by the global Galerkin method. The present method is applieddirectly to the von Kármán equations to derive a system of cubic order fully coupled NAEs with as manyunknowns as desired. Then the resulting system of NAEs is solved by an algebraic equation solver, suchas the Newton–Raphson method. Previously, Dai, Paik and Atluri [Dai et al. 2011a] applied the globalGalerkin method to the von Kármán plate, and derived the Galerkin-resulting system of NAEs explicitly.However, the explicit expression for Jacobian matrix (the so-called “tangent-stiffness” matrix) was notobtained, and should be calculated symbolically at each step. As a contribution of the present study,we derive the explicit expression of the Jacobian matrix (or the tangent stiffness matrix) of the resultingNAEs, so that the iterative methods, which require the numerical inversion of the Jacobian matrix, maybe applied more efficiently. Eliminating the symbolic operations makes the computational efficiencymuch more improved than that in [Dai et al. 2011a], as will be verified in numerical illustrations.

The global Galerkin method is compared with the incremental Galerkin method, the perturbationmethod, the finite element method and the finite difference method in solving the von Kármán plateequations under a combination of in-plane and out-of-plane loads to test its accuracy and efficiency. Inaddition, the buckling behavior and jump phenomenon of the plate are detected and analyzed numerically.

Another topic of this study is to review a recently proposed class of scalar homotopy methods forsolving NAEs. Conventionally, the Newton–Raphson method is popularly used to find successivelybetter approximations to the solutions of a real valued nonlinear system. The Newton–Raphson methodconverges remarkably quickly provided that the initial guess is sufficiently close to the solution. However,the Newton–Raphson method in general requires the inversion of the Jacobian matrix in each iterativestep, it is sensitive to the initial guess, and the accuracy of the solution cannot be guaranteed if nearlysingular or ill-conditioned Jacobian matrix is encountered. When the Jacobian (tangent stiffness) ma-trix becomes singular, as in limit load problems, researchers over the past three decades have devisedenhancements to the Newton–Raphson method, such as the arclength method. In this paper we presentmore elegant algorithms which do not involve the inversion of the Jacobian and which are simpler to usewhen the Jacobian is nearly singular.

We introduce a series of scalar homotopy methods based on the Newton scalar homotopy function.The general form of the Newton homotopy methods is used to incorporate all the existing homotopymethods in a uniform framework. Besides, the present paper provides a concise and clear interpretationfor the scalar homotopy methods, and the efficiencies of the various methods are tested through usingthem to solve the Galerkin-resulting system of NAEs. The presented scalar homotopy methods overcomethe several known drawbacks of the Newton–Raphson method:

(1) They can be applied more efficiently than the Newton–Raphson method, when the unknown vectorto be solved from the NAEs tends to be large (even through we have limited our study in this paperto only 40 nonlinear algebraic equations).

(2) They completely avoid the need for the inversion of the Jacobian matrix either numerically or ana-lytically (which is impossible in most cases).

(3) They perform much better than the Newton–Raphson method, when the Jacobian matrix is nearlysingular, or is severely ill-conditioned.


0w

yM

yP

a

yM

yP

y( , )Q x y

( , )Q x y

x

xPxMxP xMb

Figure 1. A rectangular plate with initial deflection and general in-plane and lateral loads.

(4) They are insensitive to the guess of the initial solution vector, unlike the Newton–Raphson method.

(5) They can solve either over-determined or under-determined systems of NAEs.

The paper is organized as follows. In Section 2, the global Galerkin method is used to transform thegoverning PDEs into a system of NAEs. The explicit form of the Jacobian matrix of the NAEs is alsoderived. In Section 3, various NAE solvers are introduced. A series of Newton homotopy methods areillustrated and classified into continuous-type Newton homotopy and iterative-type Newton homotopymethods. Consequently all the existing Newton homotopy methods are incorporated into a correspond-ing uniform framework. Moreover, features of the solvers are discussed. We note that Section 3 canstand alone for researchers who are interested in the new NAE solvers. Researchers who focus on thesemianalytical methods or nonlinear behavior of the plate may skip this part. Numerical experiments arecarried out in Section 4. Finally, we draw some conclusions about the present global Galerkin methodand the NAE solvers in Section 5.

2. Governing equations and the Galerkin method

The elastic large deflection response of a plate with initial deflection is governed by two PDEs, whichare named von Kármán equations. One of them represents the equilibrium condition in the transversedirection, and the other represents the compatibility condition of in-plane strains. The PDEs are asfollows:

φ ≡ D∇4w− t[ϕ,yy(w+w0),xx +ϕ,xx(w+w0),yy − 2ϕ,xy(w+w0),xy

]− Q = 0, (1a)

∇4ϕ = E

[w2,xy −w,xxw,yy + 2w0,xyw,xy −w0,xxw,yy −w0,yyw,xx

]. (1b)

In the above, w0 is the given initial transverse displacement; w is the additional transverse displacement;Q is the lateral pressure acting on the plate; ϕ is the Airy stress function governing the in plane stress


resultants and t is the plate thickness.

∇4=∂4

∂x4 + 2∂4

∂x2∂y2 +∂4

∂y4 , D =Et3

12(1− ν)2, (2)

where D is the flexural rigidity, and ∇4 is the well known biharmonic operator; E and ν are Young’smodulus and Poisson’s ratio. The subscripts ,x and ,y stand for ∂/∂x and ∂/∂y.

Stress σx in the x direction, σy in the y direction and shear stress τxy in xy plane may be expressed as

σx = ϕ,yy, σy = ϕ,xx , τxy = ϕ,xy .

We emphasize that, in using the Galerkin method, we need to solve the Airy stress function ϕ firstfrom the (1b), and then apply the Galerkin approach to (1a), which is different from the Rayleigh–Ritzmethod where the airy stress function ϕ and even the governing equations are not required. Rayleigh–Ritz method based on Lagrangian equations is simple in application but computationally expensive sincemore freedoms are expected by Rayleigh–Ritz method than by Galerkin method, because the deflectionsof all three directions are required to be assumed. In this study, we derive the explicit expressions of theϕ and then the resulting NAEs, so that researchers can avoid lengthy algebra and enjoy the advantage ofthe Galerkin method. The geometry and general loading conditions of the plate is plotted in Figure 1.

In solving the governing equations, the added deflection w due to the applied load, and the initialdeflection w0 should satisfy the boundary conditions at four edges. All four edges are assumed to besimply supported, and the boundary conditions of the plate are

w = 0, w,yy + νw,xx= 0, at y = 0 and y = b,

w = 0, w,xx + νw,yy= 0, at x = 0 and x = a.

2.1. Application of the Galerkin method. To satisfy the boundary conditions, the initial deflection w0

and the added deflection function w can be normally assumed in double Fourier series,

w0 =

M∑m=1

N∑n=1

A0mn sinmπx

asin

nπyb, (3)

w =

M∑m=1

N∑n=1

Amn sinmπx

asin

nπyb, (4)

where A0mn and Amn are the known and unknown coefficients, respectively. The present simply supportedplate can be solved with various patterns of external loads. The conditions of the combined loads, namely,biaxial loads, biaxial in-plane bending and edge shear are given as follows:∫ b

0ϕ,yy t dy = Px ,

∫ b

0ϕ,yy t

(y− b

2

)dy = Mx at x = 0 and x = a, (5a)∫ a

0ϕ,xx t dx = Py,

∫ a

0ϕ,xx t

(x − a

2

)dx = My at y = 0 and y = b, (5b)

ϕ,xy =−τ at all four edges. (5c)


Then the homogenous solution ϕh for the Airy stress function ϕ should satisfy the condition of thecombined loads acting on the plate. Considering the loading conditions, we can easily find ϕh , byassuming ϕh as cubic polynomials in x and y. Substituting ϕh into (5) we can obtain,

ϕh =−Pxy2

2bt− σr x

y2

2− Py

x2

2at− σr y

x2

2−Mx

y2(2y− 3b)b3t

−Myx2(2x − 3a)

a3t− τxy xy. (6)

The following notations are introduced to abbreviate the expressions involving the sine or cosine terms:

sinmπx

a≡ sx(m), cos

mπxa≡ cx(m), sin

nπyb≡ sy(n), cos

nπyb≡ cy(n).

To find the particular solution ϕp that should satisfy (1b), one ought to substitute w and w0 into the rightside of (1b), thus obtaining

∇4ϕp =

Eπ4

4a2b2

M∑m=1

N∑n=1

K∑k=1

L∑l=1

{[Amn Aklml(nk−ml)− Akl A0mn(nk−ml)2

]cx(m+k)cy(n+ l)

+[Amn Aklml(nk+ml)+ Akl A0mn(nk+ml)2

]cx(m+k)cy(n− l)

+[Amn Aklml(nk+ml)+ Akl A0mn(nk+ml)2

]cx(m−k)cy(n+ l)

+[Amn Aklml(nk−ml)− Akl A0mn(nk−ml)2

]cx(m−k)cy(n− l)

}. (7)

Consequently, motivated by the form of the right-hand side of (7), the particular solution ϕp for the Airystress function is assumed as

ϕp =Eπ4

4a2b2

M∑m=1

N∑n=1

K∑k=1

L∑l=1

{B1(m,n,k, l)cx(m+k)cy(n+l)+B2(m,n,k, l)cx(m+k)cy(n−l)

+B3(m,n,k, l)cx(m−k)cy(n+l)+B4(m,n,k, l)cx(m−k)cy(n−l)}. (8)

Upon substituting (8) into (1b), the coefficients Bi , i = 1, 2, 3, 4 can be readily calculated; they are notwritten out for saving space. Then, substituting the Bi into (8), we obtain the particular solution ϕp:

ϕp =Eα2

4

M∑m=1

N∑n=1

K∑k=1

L∑l=1

{Amn Aklml(nk−ml)− Akl A0mn(nk−ml)2

[(m+ k)2+ (n+ l)2]2cx(m+ k)cy(n+ l)

+Amn Aklml(nk+ml)+ Akl A0mn(nk+ml)2

[(m+ k)2+ (n− l)2]2cx(m+ k)cy(n− l)

+Amn Aklml(nk+ml)+ Akl A0mn(nk+ml)2

[(m− k)2+ (n+ l)2]2cx(m− k)cy(n+ l)

+Amn Aklml(nk−ml)− Akl A0mn(nk−ml)2

[(m− k)2+ (n− l)2]2cx(m− k)cy(n− l)

}. (9)

Then, the Airy stress function ϕ can be expressed as

ϕ = ϕh +ϕp. (10)

It is evident from Equations (6), (9) and (10) that ϕ is a second-order function with regard to theunknown deflection coefficients Amn . To compute the unknown coefficients, the global Galerkin method


is applied to the equilibrium (1a):∫∫∫Vφ(x, y, z)sx(i)sy( j) dx dy dz = 0. (11)

Upon substituting (10) into (1a), and then (1a) to (11) after a lengthy derivation, we obtain a system ofthird-order (cubic) coupled NAEs, with respect to the unknown coefficients, the explicit expression ofthe nonlinear system of algebraic equations is

∑∑Amn · Dπ4

(m2

a2 +n2

b2

)2

H01(i, j,m, n)

+

∑∑∑∑∑∑Amn Akl Ars ·(−t)

Eα2π4

4a2b2 (H1+H2+H3+H4−2H9−2H10−2H11−2H12)

+

∑∑∑∑Amn Akl ·(−t)

Eα2π4

4a2b2

∑∑A0rs(H1+H2+H3+H4−2H9−2H10−2H11−2H12)

+

∑∑∑∑Akl Ars ·(−t)

Eα2π4

4a2b2

∑∑A0mn(H6+H7−H5−H8+2H13−2H14−2H15+2H16)

+

∑∑Akl ·(−t)

Eα2π4

4a2b2

∑∑∑∑A0mn A0rs(H6+H7−H5−H8+2H13−2H14−2H15+2H16)

+

∑∑Amn · (−t)

{m2π2

a2

[(Px

bt+ σr x −

6b2t

Mx

)H01(i, j,m, n)+

12b3t

Mx H03(i, j,m, n)]

+n2π2

b2

[(Py

at+σr y−

6a2t

My

)H01(i, j,m, n)+

12a3t

My H02(i, j,m, n)]

+2τπ2

abmn · H04(i, j,m, n)

}

+

∑∑A0mn · (−t)

{m2π2

a2

[(Px

bt+ σr x −

6b2t

Mx

)H01(i, j,m, n)+

12b3t

Mx H03(i, j,m, n)]

+n2π2

b2

[(Py

at+σr y−

6a2t

My

)H01(i, j,m, n)+

12a3t

My H02(i, j,m, n)]

+2τπ2

abmn · H04(i, j,m, n)

}−Q · H00(i, j)= 0, (12)

where for simplicity the coefficient matrix H1(i, j,m, n, k, l, r, s) is denoted by H1 and so forth, and allthe summations above are carried out over the dummy indexes m, n, k, l, r, s, and i, j are free indexes.All the coefficient matrices can be obtained by performing integration over the whole volume of theplate, whose expressions are provided in the Appendix of [Dai et al. 2011a]. We can write the resulting


algebraic system (12) neatly in matrix form:

Kt At + Ks As + K f Af +C = 0, (13)

where C is a constant column matrix, K f , Ks and Kt are the first-, second- and third-order coefficientmatrices — the dimensions (number of rows × number of columns) of these matrices being

M N × (M N )3 for Kt , M N × (M N )2 for Ks, M N ×M N for K f , M N × 1 for C,

— and where Af , As and At are the first-, second- and third-order unknown vectors.1

2.2. An explicit derivation of the Jacobian matrix (tangent stiffness matrix) for the von Kármán plate.Up to now, the NAEs are obtained, and can be solved by applying NAE solvers. However, in most of thecases, the Jacobian matrix of the derived system of NAEs is required to enable the iterative proceduresin the course of using the algebraic equation solvers. In our previous work, we only derived the explicitform of algebraic system. The Jacobian matrix to this system is not derived explicitly. Therefore, weresort to symbolic operations embedded in the Matlab to calculate the Jacobian matrix at each iteration.Although very accurate solutions were achieved in [Dai et al. 2011a] via this scheme, we admit that thecomputational efforts are very heavy. In the present work, we derive the explicit form of the Jacobianmatrix to eliminate this drawback. It should be emphasized that the explicit form of the Jacobian matrixof the NAEs resulting from the von Kármán plate PDEs is provided for the first time in literature. ThisJacobian matrix needs to be inverted in each iterative step in the Newton–Raphson method, but such aninversion is not necessary in any of the scalar homotopy methods presented in this paper.

The resulting algebraic system (13) can be written in a general form as

F(A)= 0, (14)

or, particularlyFi j (Apq)= 0, i, p = 1, 2, . . . ,M; j, q = 1, 2, . . . , N . (15)

It should be noted that Fi j does not represent a matrix. It is used to simply denote that there are M × Nequations, with the [( j − 1)N + i]-th equation being denoted by Fi j . Similarly, Apq represents the[(q − 1)N + p]-th unknown coefficient.

Bu,v =∂Fi j

∂Apq, (16)

where the Bu,v is the u-th row and v-th column of B with u = ( j − 1)N + i , v = (q − 1)N + p. Thereare eight terms in (12), the first six terms are associated with the unknown coefficients and the last twoterms are constant with regard to unknowns. The Jacobian matrix B of (12) is derived term by term asfollows.

∂F1i j

∂Apq= Dπ4

(p2

a2 +q2

b2

)2

H01(i, j, p, q).

1The second- and third-order unknown vectors are vectors devised in [Dai et al. 2011a; 2011b] for obtaining the matrixequation (13).


∂F2i j

∂Apq=

∑∑∑∑Akl Ars × (−t)

Eα2π4

4a2b2 (H1+ H2+ H3+ H4− 2H9− 2H10− 2H11− 2H12)

+

∑∑∑∑Amn Ars × (−t)

Eα2π4

4a2b2 (H1+ H2+ H3+ H4− 2H9− 2H10− 2H11− 2H12)

+

∑∑∑∑Amn Akl × (−t)

Eα2π4

4a2b2 (H1+ H2+ H3+ H4− 2H9− 2H10− 2H11− 2H12),

the matrices Hi in the first, second and third line being evaluated, respectively, as follows:

H(i, j, p, q, k, l, r, s), H(i, j,m, n, p, q, r, s), H(i, j,m, n, k, l, p, q).

∂F3i j

∂Apq=

∑∑Akl × (−t)

Eα2π4

4a2b2

∑∑A0rs(H1+ H2+ H3+ H4− 2H9− 2H10− 2H11− 2H12)

+

∑∑Amn × (−t)

Eα2π4

4a2b2

∑∑A0rs(H1+ H2+ H3+ H4− 2H9− 2H10− 2H11− 2H12),

where the matrices Hi in the first and second lines are H(i, j, p, q, k, l, r, s) and H(i, j,m, n, p, q, r, s),respectively.

∂F4i j

∂Apq=

∑∑Ars × (−t)

Eα2π4

4a2b2

∑∑A0mn(H6+ H7− H5− H8+ 2H13− 2H14− 2H15+ 2H16)

+

∑∑Akl × (−t)

Eα2π4

4a2b2

∑∑A0mn(H6+ H7− H5− H8+ 2H13− 2H14− 2H15+ 2H16),

where the matrices Hs in the first and second line are H(i, j,m, n, p, q, r, s) and H(i, j,m, n, k, l, p, q),respectively.

∂F5i j

∂Apq= (−t)

Eα2π4

4a2b2

∑∑∑∑A0mn A0rs(H6+ H7− H5− H8+ 2H13− 2H14− 2H15+ 2H16).

∂F6i j

∂Apq= (−t)

{p2π2

a2

[(Px

bt+ σr x −

6b2t

Mx

)H01+

12b3t

Mx H03

]

+q2π2

b2

[(Py

at+ σr y −

6a2t

My

)H01+

12a3t

My H02

]+

2τπ2

abpq H04

},

where the matrices depend on (i, j, p, q). Therefore, the Jacobian matrix is

Bu,v =∂Fi j

∂Apq=

6∑k=1

∂Fki j

∂Apq. (17)

Consequently, with the explicit form of the system of NAEs and its Jacobian matrix, the present problemcan be solved readily by using various NAE solvers. In the course of solving a system of NAEs, theJacobian matrix is usually necessary. Normally, the numerical approximation of the Jacobian matrix iscalculated in each iteration step via numerical difference techniques. The explicitly derived Jacobian


matrix may significantly accelerate the computing rate of the algebraic solver [Dai et al. 2014a]. Theeffect of using the explicitly derived Jacobian matrix rather than the numerically calculated one on thecomputational efficiency has been intensively analyzed in [Dai et al. 2014b] in a two-degree-of-freedomairfoil problem. It was demonstrated that using the explicit Jacobian matrix can be roughly two ordersof magnitude faster.

3. Methods for nonlinear algebraic equations

The numerical solution of linear or nonlinear, well-conditioned or ill-conditioned, and underdeterminedor overdetermined algebraic equations is one of the main aspects of computational mechanics. In manypractical nonlinear engineering problems, methods such as the finite element method, boundary elementmethod, finite volume method, the meshless method, global Galerkin method, Rayleigh–Ritz method,etc., eventually lead to a system of nonlinear algebraic equations (NAEs). Many numerical methodsused in computational mechanics, as illustrated in [Atluri 2005] lead to the solution of a system of linearalgebraic equations for a linear problem, and of a system of NAEs for a nonlinear problem.

A system of nonlinear algebraic equations is

Fi (x j )= 0, i, j = 1, 2, . . . , n. (18)

Solvers for this set of NAEs are introduced below. In the section of numerical experiments, they areapplied to solve the resulting algebraic system from the implementation of the Galerkin method tovon Kármán plate equations.

3.1. Newton method and preliminary work. The most famous method for solving nonlinear algebraicequations is the Newton–Raphson method, or Newton method, which is given algorithmically as

xk+1 = xk − B−1k Fk, (19)

where we use x := x1, x2, . . . , xn and F := F1, F2, . . . , Fn to represent the vectors, B is the n × nJacobian matrix with its (i, j) entry given by ∂Fi/∂x j , and xk+1 is the (k+1)-th iteration for the unknownvector x. Newton method is advantageous in that it converges quadratically fast, provided that the initial

“guesses” for the solution are within a certain radius of convergence. However, sometimes Newtonmethod suffers from its sensitiveness to initial “guesses”, and the computational burden/accuracy ofinverting the Jacobian matrix when the Jacobian matrix is singular or severely ill-conditioned.

Hirsch and Smale [1979] derived a “continuous Newton method” governed by the differential equation

x(t)=−B−1 F(x), (20)

x(0)= a, (21)

where a ∈ Rn . It should be noted that applying a forward Euler scheme to (20) leads to the classicalNewton method. Therefore, the continuous Newton method is just the iterative form of Newton methodwritten in the ODE form. The performance does not improve much as compared with the classicalNewton method.

Until very recently Newton-type methods are the only choice for solving NAEs, where the inverse ofJacobian matrix is inevitable. To eliminate the need for inverting a matrix in the iteration procedure, a


straightforward first-order ODE system,

x(t)=−F(x), (22)

x(0)= a, (23)

was used [Ramm 2004]. However, iteration procedure arising out of the integration of (22) is verysensitive to the initial guess, and converges very slowly. Liu and Atluri [2008] proposed a fictitious timeintegration method (FTIM) in the form of a system of ODEs as

x(t)=−ν

q(t)F(x), (24)

where ν is a nonzero parameter and q(t) is required to be a monotonically increasing function of t . In theirapproach, the term ν/q(t) plays the role of speeding up the convergence. It is noted that an elementaryversion of the continuation method similar to the FTIM was introduced in [Kane and Levinson 1985].

However, both the methods of FTIM and that in [Ramm 2004] are not rigorously derived mathemati-cally. Interestingly, we can see that both of them do not need the Jacobian matrix, let alone its inversion.However, according to authors’ experience, they are extremely sensitive to initial guesses, and convergemuch more slowly than the Newton method. Therefore both methods are not recommended unless theJacobian matrix cannot be obtained or involved.

Atluri, Liu and Kuo [Atluri et al. 2009] proposed a modified Newton method (MNM), which is in facta combination of the continuous Newton method and the FTIM and the finite difference technique, forsolving nonlinear algebraic equations avoiding the inverse of the Jacobin matrix. The MNM is given as

dxi

dτ=−

ν

1+ τ(1− si )Bi

xi − xi−1

1s+ Fi = 0, i = 1, . . . ,m, (25)

where s = 1 − e−t is a new variable, and s ∈ [0, 1) is divided into m subintervals with 1s = 1/m.Numerical examples of [Atluri et al. 2009] showed that the MNM converges faster than the FTIM insome problems. However, the convergence rate still cannot compare with that of the Newton method. Inaddition, the ODE system (25) of the MNM is m times larger than the FTIM and the continuous Newtonmethod, which makes the integration much more expensive.

Liu, Yeih, Kuo and Atluri [Liu et al. 2009] developed a scalar homotopy method, which transforms theoriginal NAEs into an equivalent system of ODEs. The scalar homotopy method is totally distinguishedfrom the aforementioned FTIM, MNM methods because the FTIM and MNM are based on (24), whichis not a strictly derived relation but rather an intuition.

In solving nonlinear algebraic equations, the homotopy method represents a way to enhance theconvergence from a local convergence to a global convergence. Previously, all the homotopy methodsare based on the construction of a vector homotopy function, H(x, t) which serves the objective ofcontinuously transforming a function G(x) into F(x) by introducing a homotopy parameter t . Thehomotopy parameter t can be treated as a time-like fictitious variable, and the homotopy function can beany continuous function such that: H(x, 0)= G(x) and H(x, 1)= F(x).

Two kinds of homotopy functions are popularly used. The fixed-point homotopy function can bewritten as

H(x, t)= t F(x)+ (1− t)(x− x0)= 0, (26)


and the Newton homotopy function is

H(x, t)= t F(x)+ (1− t)[F(x)− F(x0)] = 0, (27)

where x0 is the given initial values and t ∈ [0, 1]. Motivated by the above vector homotopy function (26),Liu et al. [2009] proposed a fixed-point scalar homotopy function

h(x, t)= 12 t‖F(x)‖2+ 1

2(t − 1)‖x− x0‖2= 0, (28)

Then x = x(t) is assumed in [Liu et al. 2009]. Differentiating (28) on both sides with respect to t yields

12 [‖F(x)‖2+‖x− a‖2] + [t BT F− (1− t)(x− a)] · x = 0. (29)

Also, x needs to be parallel to the gradient of the above scalar homotopy function, such that the trajectoryof x can be equivalent to seeking of h(x, t)= 0. Thus,

x =−λ∂h∂x. (30)

Therefore, using Equations (28)–(30) the scalar homotopy method (SHM) is derived as

x =−12‖F(x)‖2+‖x− a‖2

‖t BT F− (1− t)(x− a)‖2[t BT F− (1− t)(x− a)]. (31)

The SHM is the first scalar homotopy method, and it is based on the fixed point scalar homotopy function.This method is proved to be less sensitive to initial guess, it has an acceptable convergence rate [Liuet al. 2009]. Systems of over/under determined algebraic equations, or systems being sensitive to initialguesses, or systems whose Jacobian matrix is ill-conditioned, can be solved by the SHM method betterthan by the Newton method.

3.2. Continuous Newton homotopy methods. In this study, we introduce a series of continuous algo-rithms based on the Newton homotopy function. The general form of the Newton homotopy methodsincorporates all the existing homotopy methods in a uniform framework.

The Newton homotopy function (27) can be written as

H(x, t)= F(x)+ (t − 1)F(x0)= 0. (32)

Similar to the process in SHM, we can transform the vector Newton homotopy function into a scalarform as follows:

h(x, t)= 12‖F(x)‖2+ 1

2(t − 1)‖F(x0)‖2= 0. (33)

Equation (33) holds for all t ∈ [0, 1]. To motivate this study, we first consider a fictitious time functionQ(t), t ∈ [0,∞), where t is the fictitious time and Q(t) has to satisfy that Q(t) > 0, Q(0)= 1, and Q(t)is a monotonically increasing function of t , and Q(∞)=∞. Then we introduce the proposed fictitioustime function Q(t) into (33) and have a generalized scalar Newton homotopy function

h(x, t)= 12‖F(x)‖2− 1

2Q(t)‖F(x0)‖

2= 0, (34)


Using the fictitious time function, Q(t), when the fictitious time t = 0 and t =∞, we can obtain

h(x, t = 0)= 12‖F(x)‖2− 1

2‖F(x0)‖2= 0⇔ F(x)= F(x0), (35)

h(x, t =∞)= 12‖F(x)‖2 = 0⇔ F(x)= 0. (36)

Clearly, the tracking of a solution path for the proposed scalar Newton homotopy function is equivalentto the fictitious time varying from zero to infinity. Multiplying both sides of (34) by Q(t) we have

h(x, t)= 12 Q(t)‖F(x)‖2− 1

2‖F(x0)‖2= 0, (37)

We expect h(x, t) to be an invariant manifold in the space of (x, t) for a dynamical system h(x(t), t)to be specified further. With the assumption of Q(t) > 0, the manifold defined is continuous, and thus thefollowing operation of differential carried out on the manifold makes sense. As a consistency condition,by taking the time differential of (37) with respect to t and considering x = x(t), we have

12 Q(t)‖F(x)‖2+ Q(t)(BT F) · x = 0. (38)

To transform the original NAEs to ODEs, x should be specified like x = λu. It should be emphasizedthat there are a variety of choices for the form of x = λu. Various Newton homotopy methods maygenerate from selections of u. Initially, we assume

x = λu. (39)

Substituting (39) into (38) yields

λ=−Q(t)

2Q(t)‖F(x)‖2

FT Bu, (40)

where, λ is a scalar. Plugging λ into (39), we have

x =−Q(t)

2Q(t)‖F(x)‖2

FT Buu, (41)

Equation (41) is the general form equation for the continuous Newton homotopy methods. A classof continuous Newton homotopy methods can be obtained from this general equation by reasonablychoosing different driving vector u. It is found that a fictitious time function is introduced in (41) whichis a mathematically equivalent n (if t is implicit) or n+ 1 (if t is explicit) dimensional dynamical systemto the original algebraic equation system. The solution for the original algebraic equation can be obtainedby applying numerical integration to the equivalent dynamical ODEs.

The fictitious time function Q(t) should be specified before applying the numerical integration. Q(t),as discussed above, should be a monotonically increasing function of t . There are many choices for Q(t).According to [Ku et al. 2009], we can choose

Q(t)= eν

1−m [(1+t)1−m−1], (42)

so thatQ(t)Q(t)

=ν

(1+ t)m, 0< m ≤ 1. (43)

We make (42) the first choice of Q(t). A simpler, intuitive choice of the fictitious time function is


method driving vector u Q(t)

DNM1 B−1 F choice 1DNM2 B−1 F choice 2*MBECA1 BT F choice 1MBECA2 BT F choice 2DJIFM1 F choice 1DJIFM2 F choice 2

Table 1. A summary of continuous Newton homotopy methods.

Q(t)= et, (44)

which makes Q(t)/Q(t)= 1. This is labeled as the second choice of Q(t).Finally the general form continuous Newton homotopy method has been derived as (41) with a spec-

ified Q(t) in (42) or (44). Integrating this system of ODEs, one can arrive at the stable solution of theODEs, which is the solution of the original nonlinear algebraic system.

Different choices of the driving vector u in the general Equation (41) lead to different kinds of con-tinuous Newton homotopy methods. See Table 1 for a summary of methods.

Interestingly, if we choose Q(t)= e2t , that is, choice 2*, instead of Q = et for the DNM2, we obtain

x =−B−1 F, (45)

which turns out to be exactly the continuous Newton method by Hirsch and Smale [1979]. Applying theforward Euler scheme to (45), we have

xk+1 = xk − B−1k Fk, (46)

which is the classical Newton method.It can be seen from Table 1 that DNMs are different from the other continuous Newton homotopy

methods in that the inverse of the Jacobian matrix is involved. The DNM2 and DNM1 should be regardedas the Newton method and a variant Newton method respectively. However, the DNMs [Ku et al. 2011;Ku and Yeih 2012] are more flexible than the Newton method, since the dynamical system of the DNMscan be with different choices of Q(t) and numerical integration methods, while the Newton method is aspecial case with Q(t)= e2t and the forward Euler method. It is expected that proper selections of Q(t)and integration method may improve the convergence performance.

The MBECAs and the DJIFMs do not involve the inversion of Jacobian matrix. The MBECA1 turnsout to be exactly the same as the ECSHA, which is applied in [Dai et al. 2011a].

There are three types of continuous Newton homotopy methods as introduced above. All the threemethods are based on the driving vector u where there is only one vector in u. To be extended, we canassume u to be constructed by two vectors, such as F and BT F, or F and its normal vector P , or BT Fand its normal vector P∗. To derive the new methods, the only thing to do is to replace the u in (41). Inthis study, unless otherwise specified, we use the forward Euler method to perform the integration for thecontinuous Newton homotopy methods. The performance of the continuous Newton homotopy methodsis tested in numerical examples.


3.3. Iterative Newton homotopy methods. Subsequently, Liu and his coworkers developed a series ofpurely iterative Newton homotopy methods, where Q(t) no longer needs to be specified. Similar to thecontinuous Newton homotopy methods, these iterative Newton homotopy methods can also be incorpo-rated into a uniform framework.

To derive the purely iterative methods, the general equation (41) of the continuous Newton homotopymethods is first discretized into a discrete time dynamics via the forward Euler method:

x(t +1t)= x(t)−β‖F(x)‖2

FT Buu, (47)

where

β = q(t)1t and q(t)=Q(t)

2Q(t). (48)

Then, we differentiate F with respect to t , and obtain

F = Bx =−q(t)‖F‖2

‖BT F‖2AF, (49)

where A= B BT . Similarly, we use the forward Euler scheme to integrate (49) and get

F(t +1t)= F(t)−β‖F(x)‖2

FT Buu. (50)

Considering that formula (37) is an invariant manifold in time and letting C = 12‖F(x0)‖

2, we can get

‖F(t)‖2 =2C

Q(t), (51)

‖F(t +1t)‖2 =2C

Q(t +1t), (52)

since the defined manifold should be invariant with time. Squaring both sides of (50) and using Equa-tions (51) and (52) we can obtain

CQ(t +1t)

=C

Q(t)− 2β

CQ(t)+β2 C

Q(t)‖F‖2

(FT Bu)2‖Bu‖2. (53)

After some simple algebra, the following scalar equation is obtained:

a0β2− 2β + 1− s = 0, (54)

where

a0 :=‖F‖2‖Bu‖2

‖FT Bu‖2, s =

Q(t)Q(t +1t)

=‖F(t +1t)‖2

‖F(t)‖2. (55)

It worth noting that s can be used as a quantity to assess the convergence property of the iterative Newtonhomotopy methods, and a0 ≥ 1 according to the Cauchy–Schwarz inequality

‖F · (Bu)‖ ≤ ‖F‖‖Bu‖. (56)


From (54), we can take the solution of β to be

β =1−√

1− (1− s)a0

a0. (57)

To ensure 1− (1− s)a0 ≥ 0, let

1− (1− s)a0 = γ2≥ 0, (58)

s = 1−1− γ 2

a0, (59)

and from (57) it follows that

β =1− γ

a0. (60)

From Equations (47), (55) and (60) we can obtain the algorithm

x(t +1t)= x(t)− (1− γ )FT Bu‖Bu‖2

u, (61)

where−1< γ < 1 (62)

is a parameter to be chosen by user. Equation (61) is the general form of the iterative Newton homotopymethods.

Using Equations (55), (59) and (62) we derive that

‖F(t +1t)‖‖F(t)‖

=√

s < 1, (63)

which means that the residual error is absolutely decreased. This property guarantees that the algorithmin (61) is absolutely convergent to the true solution, and a smaller s implies a faster convergence rate.

Recall that the continuous Newton homotopy methods involve the fictitious time function, and thedynamical system (41) should be integrated in time, step by step. Conversely, the iterative Newtonhomotopy methods are purely iterative, and do not require a specification of Q(t). Different choicesof the driving vector u in the general Equation (61) will lead to different iterative Newton homotopymethods as summarized in Table 2, wherein R = BT F, C = BT B.

The RNBA is the first iterative Newton homotopy method, which employs one vector R in the drivingvector u. Later, a series of iterative Newton homotopy methods employing two vectors in the drivingvector are developed. The OVDA uses u= αF+βR, and the dividing parameters α and β are determinedby letting ∂s/∂α = 0 and β = 1−α.

Liu, Dai and Atluri [2011a; 2011b] proposed the OIA/ODVs and the OIAs. The main differencebetween the OIA/ODVs, OIAs and the OVDA is that two orthogonal vectors instead of the couple ofF and R are used to constitute the driving vector. Numerical examples in [Liu et al. 2011a; 2011b]illustrated that OIA/ODVs and OIAs have a better performance than the OVDA in solving their selectedproblems, while this is not always the case. Numerical examples of this study indicate that the OIAs andthe OVDA are comparable in terms of convergence rate, while the OIA/ODVs converge more slowly.In particular, the OIA/ODV[R] is several times slower than the OIAs and the OVDA. So the OIAs arebelieved to be superior to the OIA/ODVs.


method driving vector u P (or P∗) parameter optimization scheme

RNBA [Liu and Atluri 2011b] R * *

OVDA [Liu and Atluri 2011a] αF+βR * ∂s/∂α = 0, β = 1−α

OIA/ODV[R] [Liu et al. 2011a] αR+β P R− ‖R‖2RT C R

C R ∂s/∂α = ∂s/∂β = 0

OIA/ODV[F] [Liu et al. 2011a] αF+β P∗ F− ‖F‖2FT C F

C F ∂s/∂α = ∂s/∂β = 0

OIA(R) [Liu et al. 2011b] αR+β P F− R·F‖R‖2

R ∂s/∂α = ∂s/∂β = 0

OIA(F) [Liu et al. 2011b] αF+β P∗ R− R·F‖F‖2

F ∂s/∂α = ∂s/∂β = 0

LOIA [Liu and Atluri 2012] αF+ R * ∂s/∂α = 0, β = 1−α

GOIA [Liu and Atluri 2012] αF+ R * global minimum of s

Table 2. A summary of iterative Newton homotopy methods.

It can be seen from Table 2 that the LOIA is essentially similar to the OVDA method, so only theOVDA is evaluated via numerical experiments. For more details about the Newton homotopy methods,one is recommended to refer to related references in Table 2.

4. Numerical examples

In this section, examples concerning the solution of von Kármán nonlinear plate equations, for a plateundergoing various kinds of loads are provided to verify the present method as well as to evaluate thenovel algebraic equation solvers. The Young’s modulus and Poisson’s ratio are E = 205.8× 109 andν = 0.3, unless otherwise specified. For NAE solvers, the parameter γ = 0.3 and the stop criterionε = 10−4 are fixed throughout the paper.

In some examples, the external loads are applied in terms of critical values Pxcr or τcr. The criticalvalues of Pxcr and τcr for a plate depend on its supporting pattern as well as length versus width ratio.For a simply supported plate with a/b = 1 or 2, the critical values are given as follows

σxcr = 4×π2 Db2t

,ab= 1, 2,

τcr = k×π2 Db2t

, k = 9.34 ifab= 1; k = 6.6 if

ab= 2,

where σxcr = Pxcr/(bt). For more types of supporting forms and length/width ratios, one may refer toChapter 9 of [Timoshenko 1961].

4.1. A square plate under uniaxial compression. In this example, a simply supported square plate underuniaxial compression is analyzed. The dimensions of this plate are a = 1, b = 1, t = 0.009, where a, b, trepresent length, width and thickness respectively. All dimensions in this study are in meters unless


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Maximum Deflection / Thickness

UniaxialCompression

(Px

Pxcr)

1× 1 Present Method


Incremental Galerkin Method [Ueda, Rashed and Paik (1987)]

Figure 2. Deflection versus compression loads.

otherwise mentioned. The initial deflection is specified as A0mn = 0.45× 10−3. Pxcr is the criticalcompression causing buckling of plate.

To examine the accuracy of the present global Galerkin method, a case of the incremental Galerkinmethod developed by Ueda, Rashed and Paik [Ueda et al. 1987] is used to compare with the presentmethod. Figure 2 plots the load-deflection relationships obtained at different load levels. The compres-sion load acting on the plate varies from 0.1 to 2 with load step being 0.1. Therefore, there are 20 loadsteps and hence 20 sets of NAEs to solve. For the first load step, the initial values are chosen as a setof small values rather freely, since the expected solution is small. In addition to the first load step, theinitial values to start the NAE solver are obtained through a load marching procedure, where the solutionof the previous load step is used as the initials of the current load step.

It can be seen from Figure 2 that the results of the present Galerkin method and the incrementalGalerkin method are in very good agreement. Figure 2 also reveals that the present method with nineterm (3 × 3) deflection function (labeled as 3 × 3 present method) agrees very well with the 1 × 1present method. In the square plate case, 1× 1 present method is quite accurate owing to the similaritybetween the one half wave plate deflection and the 1× 1 deflection function, which indicates solving thesimply supported square plate is extremely economic via the present global Galerkin method. Numericalcalculations throughout the paper are performed in Matlab on a personal computer with an Intel corei5 CPU.

4.1.1. Comparative performance of various NAE solvers. Another purpose of the present study is toevaluate the various kinds of NAE solvers. We employ the 3× 3 present method to solve the von Kármánplate problem. Thus, a system of nine NAEs is obtained. Table 3 provides the computational informationfor solving the resulting system of NAEs via various kinds of solvers. The number of iterations, comput-ing time, and time per iteration (TPI) are listed in Table 3, from which we can see that DNM2, that is,


method time(s) iterations TPI(s)

SHM 2923.9 184679 0.0158MBECA1 289.9 19144 0.0151MBECA2 859.7 56758 0.0151DNM1 1.9 76 0.0255DNM2 1.7 72 0.0233DJIFM1 26.9 1849 0.0145DJIFM2 51.8 3419 0.0152

OVDA 20.2 999 0.0202OIA/ODV[R] 317.4 16142 0.0197OIA/ODV[F] 26.5 1341 0.0198OIA(R) 20.2 1020 0.0198OIA(F) 20.1 1001 0.0201GOIA 19.6 980 0.0200

Table 3. Example 1: 3× 3 present method.

the classical Newton method, and its variant DNM1 are the fastest ones. It implies that the best drivingvector should employ the inverse of the Jacobian matrix. However, since the DNM1 and DNM2 requirecalculating the inverse of the Jacobian matrix, the TPIs of them are expected to be larger than those ofthe other methods, which is justified in Table 3. It is emphasized that the larger TPIs do not influencemuch the computational costs due to the relatively smaller size of the Jacobian matrix. However, whenthe number of NAEs is very large, the initial “guesses” of solution are not easy to generate, and whenthe Jacobian matrix is nearly singular or severely ill-conditioned, the advantages of the scalar homotopymethods such as the GOIA which does not need to invert the Jacobian matrix start becoming apparent.

The FTIM and MNM are applied to the present case; however, neither gives a convergent solution.The preliminary SHM based on the fixed-point homotopy function converges several times more slowlythan the Newton homotopy methods based on the Newton homotopy function. In general, the number ofiterations of the iterative methods of the lower part table is smaller than that of the continuous methodsof the upper part table. However the OIA/ODV[R] is verified to be an exceptional case in the family ofiterative methods, which costs ten more times iterations than its counterpart OIA/ODV[F]. The iterativeNewton homotopy methods cost less iterations than the continuous Newton homotopy methods in general.As shown in Table 3, the OVDA, the OIAs and the GOIA are comparable, and promise to be the bestJacobian-inverse-free methods; they may play an important role in solving a nonlinear problem whoseinitial guess is hard to choose, when Jacobian matrix is ill-conditioned or nearly singular, when the num-ber of NAEs tends to be very large and when the system of NAEs is overdetermined or underdetermined.

4.2. A rectangular plate under uniaxial compression. A simply supported rectangular plate under uni-axial compression is considered. Its dimensions are a = 1.68, b = 0.98 and t = 0.011. The initialdeflection is given by A011 = 1.1×10−3 and A021 = 0.22×10−3. The present Galerkin method is appliedto solve this rectangular plate. Besides, the analysis is also carried out by the FEM using rectangular,


0

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

ab

De

fle

ctio

n

Figure 3. Deformation of a rectangular plate under uniaxial compression: load = 2.

−15 −10 −5 0 5 10 150

2

4

6

8

10

12

14

16

18

Deflection (mm)

Stress

inxdirection

σx(kgf/mm

2)

Point A

Point B

A, B coincide

FEM(7× 18 mesh, 5 DOF per node)

1× 1 Present method



Figure 4. Stress σx versus deflection of point A and B, for the present Galerkin methodand the finite element method.

four node, and nonconforming plate elements with five degrees freedom at each node; 7× 18 elementsfor half of the plate.

Figure 3 provides the deformation of the rectangular plate. Figure 4 displays curves that plot the stressσx against the deflection of two points A (0.25a, 0.5b) and B (0.75a, 0.5b). It reveals that the results


of the present method and that of the FEM are in good agreement, which confirms the accuracy of thepresent method.

Comparing the solutions by the present methods in Figure 4, we see that the global Galerkin methodwith 1× 1 term deflection function cannot provide an acceptable solution, Physical intuition tells us thatthe deformation of the plate cannot be described by one term deflection function anyway. It can be seenfrom Figure 3 that the deformation of the plate has two half waves in the x direction; therefore at leasttwo terms should be used in x direction. The result of 2× 1 present method is in a very good agreementwith the result of FEM. Figure 4 also displays that solutions by 3× 2 and 2× 1 present methods areoverlapped, which indicates that present method with only a few modes can provide a very accuratesolution for the simply supported von Kármán plate under uniaxial compression.

4.2.1. Performance of solvers. The 3× 2 present method is used to solve the rectangular plate, andthe computation information for various solvers for solving the Galerkin-resultant NAEs is provided inTable 4.

In accordance with Example 1 (Table 3), the DNM1 and DNM2 involved with the inverse of theJacobian matrix have larger TPIs, while the total computing time and consumed iterations of the DNMsare much more cheap. The reason is that the usage of B−1 provides the best descent direction to reduceresiduals (hence requiring fewer iterations), and the time consumption of inverting the current Jacobianmatrix is not expensive. Although in the present case the slightly different TPIs are not sufficient toreverse the overall performance of the Newton method and the homotopy methods. It is reasonable toexpect that when the size of Jacobian is very large, the Newton method would suffer from a larger TPI.Also, the iterative Newton homotopy methods are computationally cheaper than the continuous-typemethods in general, except the OIA/ODV[R].

method time(s) iterations TPI(s)

MBECA∗ 600.7 11208 0.0536MBECA1 39.4 12555 0.0031MBECA2 63.3 20537 0.0031DNM1 0.51 109 0.0047DNM2 0.55 114 0.0048DJIFM1 4.6 1424 0.0032DJIFM2 5.9 1861 0.0032

OVDA∗ 53.2 633 0.0840OVDA 2.3 663 0.0035OIA/ODV[R] 20.4 6336 0.0032OIA/ODV[F] 2.9 866 0.0033OIA(R) 2.2 636 0.0034OIA(F) 2.3 659 0.0034GOIA 2.3 665 0.0035



0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3

Qa4/Et4

w/t

Linear theory

Ueda 1987

Levy 1942

Brown 1969

1× 1 present method


Figure 5. Square plate: central deflection versus lateral pressure.

In addition, the MBECA* [Dai et al. 2011a] (or named ECSHA) and the OVDA* [Dai et al. 2011b]using symbolic calculations of the Jacobian matrix at each iteration are also listed in Table 4. We seethat inclusion of the symbolic operations slightly improves the performance of the solvers in terms of theconsumed number of iterations, because using symbolic operations can avoid the cut-off errors whichinherently exist in numerical calculations. Nevertheless, the TPIs of MBECA* and OVDA* are aboutten times larger than those of the MBECA and OVDA, which indicates that the contribution of derivingthe explicit Jacobian matrix is of significant importance. Table 4 shows that the GOIA, the OIAs andthe OVDA are the most efficient Jacobian-inverse-free methods. Next to the above three methods arethe OIA/ODV[F] and the DJIFMs; the OIA/ODV[R] and the MBECAs are several orders of magnitudemore expensive. It is found that Newton homotopy methods having a driving vector with F inside aresuperior to those where F is not included.

4.3. A square/rectangular plate subjected to lateral load. A square plate, with geometry a = b= 1 andt = 0.009, subjected to a uniformly distributed lateral load Q is considered in this example without initialimperfections.

It indicates in Figure 5 that the present method is quite accurate in solving a plate under lateral loadthrough a comparison with the incremental Galerkin method [Ueda et al. 1987], and Levy’s method. Theresult of 1× 1 present method agrees well with that of 3× 3 present method. We conclude that for asquare plate, the present method with very few terms can be reasonably accurate, due to the similaritybetween the real plate deflection and the one half-wave bulge of the assumed function.

Figure 5 also gives the result via a finite difference method proposed by Brown and Harvey [1969].It is seen that there is a discrepancy between the present method and the finite difference method. Thedifferences between the solutions, which may be attributed to two different sets of boundary conditions,both of which might be loosely described as simply supported, indicate the importance of specifying


0 20 40 60 80 100 120 140 1600

0.5

1

1.5

2

2.5

3

Qa4/Et4

w/t

Linear theory a/b=2

Linear theory a/b=1.5

Linear theory a/b=1

present method a/b=2

present method a/b=1.5

present method a/b=1

Figure 6. Maximum deflection versus lateral pressure for a range of length/width ratios.

plate number a/b b/t E (GPa) ν Q Px

1 1 50 205.8 0.30 varies 0.60Pxcr

2 3 66.7 204.8 0.33 29400 (Pa) varies

3 3 66.7 204.8 0.33 49000 (Pa) varies

4 3 74.7 214.6 0.33 1.43( Qb4

Et4

)varies

5 3 74.7 214.6 0.33 4.28( Qb4

Et4

)varies

Table 5. Geometries and material properties of plates.

all four boundary conditions. In a further study, we are trying to replace the loose simply supportedconditions used in [Levy 1942b; Ueda et al. 1987; Paik et al. 2001; Dai et al. 2011a] by strict simplysupported conditions using the present method.

Figure 6 shows the deflection-load relationships for various length/width ratios, and the results by thelinear theory are also provided. It can be seen that increasing the ratio of length versus width will leadto a larger plate deflection, which is in accord with physical intuition. Under the same load situation, weexpect that an infinite large length/width ratio causes the largest deflection of a plate.

4.4. A square/rectangular plate subjected to lateral pressure combined with uniaxial compression.First, a square plate subjected to lateral pressure combined with uniaxial compression is considered. Thex-direction compression acting on the plate is a constant (see plate 1 in Table 5), and the lateral pressureacting on the plate increases as shown in Figure 7. The geometric and material properties are listed inTable 5, and the initial deflection is zero. For comparison purpose, this plate has also been analyzed via


0 0.5 1 1.50

5

10

15

20

25

30

Maximum Deflection / Plate Thickness

La

tera

l lo

ad

, Q

b4/E

t4

FEM


Figure 7. Square plate 1: maximum deflection versus lateral pressure by the presentmethod and the FEM.

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

3

3.5

4

w/t

Qa4/Et4

without lateral load

0.2Pcr

0.4Pcr

0.6Pcr

Figure 8. Square plate 1: maximum deflection versus lateral pressure for a range of edge compressions.

FEM in ANSYS (brick element with eight node and three degrees freedom at each node, 50× 50× 1elements for the whole volume).

It is seen from Figure 7 that the results of the present method are quite in accordance with those ofthe FEM when the lateral pressure is below approximate 15. As the lateral pressure increases further,the discrepancy between the two methods increases.


0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

w/t

compre

ssion

alongx

axis

(Pxb

4π

2D)

Experiment: Yamamoto 1970

Shen 1989

Present method

plate 2

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

w/tcompre

ssion

alongx

axis

(Pxb

4π

2D)

Shen 1989

Experiment: Yamamoto 1970

Present method

plate 3

Figure 9. Uniaxial compression versus maximum deflection with a constant lateral pres-sure by the present method, the perturbation method [Shen 1989] and the experiment[Yamamoto et al. 1970]. Axial load versus deflection: plate 2 (left) and plate 3 (right).

Figure 8 shows the deflection versus lateral pressure curves for square plate 1 under a range of edgecompressions. It indicates that as the edge loading increases the plate deflection increases accordinglyunder the same lateral pressure.

In the first case of plate 1, the axial compression is kept constant and the lateral pressure is increasedincrementally. We also analyze the cases, rectangular plates 2–5, wherein the lateral pressure is exertedfirst and kept constant, and x-direction compression is increased gradually.

Figure 9 shows the comparisons of the load-deflection relationships by the present method, the pertur-bation method [Shen 1989], and the experiment results [Yamamoto et al. 1970]. We see that the presentmethod approximately agrees with the perturbation method and the experiment. The present methodis more closer to the experiment than the perturbation method when the compression is relatively low.However, as the load increases, the deflection by the present method is smaller than that of the pertur-bation method and the experiment. The discrepancy between the present method and the experiment isapproximately a constant value, larger than the discrepancy between the perturbation method and theexperiment. The present method is simpler than the perturbation method in [Shen 1989]. Because in[Shen 1989], the Galerkin method is used to first convert lateral pressure into an initial deflection, andthen governing equations are studied using a perturbation method, taking deflection as its perturbationparameter.

Figure 10 displays the deflection-axial load relations for a range of lateral pressures. The curves arecomputed through a load-marching procedure wherein the solution of the previous load step is used asthe initial of the current load step. In Figure 10, both forward marching and backward marching pathsare plotted. It can be seen that with the increase of the lateral pressure, a hysteresis phenomenon mayoccur. In this case, Q = 3.7 has no hysteresis while Q = 4 has hysteresis; so the buckling behavior of


0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Px/P

cr

A1

/t

Q=3

0 1 20

0.5

1

1.5

2

Px/P

cr

A3

/t

2

Q=3

0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Px/P

cr

A1

/t

Q=3.7

0 1 20

0.5

1

1.5

2

Px/P

cr

A3

/t

Q=3.7

0 1 20

0.2

0.4

0.6

0.8

Px/P

cr

A1

/t

A

B

Q=4

2

B

0 1 20

0.5

1

1.5

2

Px/P

cr

A3

/t

A

B

Q=4

0 1 20

0.2

0.4

0.6

0.8

1

Px/P

cr

A1

/t

Q=5

2 0 1 20

0.5

1

1.5

2

Px/P

crA

3/t

Q=5

Figure 10. The deflection versus axial load relationships for a range of lateral loadingQs: Q = 3, 3.7, no hysteresis; Q = 4, 5, hysteresis occurs.

0 0.5 1 1.50

5

10

15

20

25

30

35

40

45

50

A1 (mm)

Px (

ton)

0 2 4 6 8 100

5

10

15

20

25

30

35

40

45

50

A3 (mm)

Px (

ton)

Okada 1979


plate 4

0 1 2 3 40

5

10

15

20

25

30

35

40

45

50

A1 (mm)

Px (

ton)

0 2 4 6 8 100

5

10

15

20

25

30

35

40

45

50

A3 (mm)

Px (

ton)

Okada 1979

present, forward

present, backward

plate 5Figure 11. Axial load versus deflection: comparison between present method and[Okada et al. 1979]: not buckling (plate 4); buckling (plate 5).


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

x

y

De

fle

ctio

n

Figure 12. Deformation of a rectangular plate under uniaxial compression Px = 0.65Pxcr

(applied first and keep constant) and shear τ = 2τcr (applied gradually).

the plate occurs at between Q = 3.7 and Q = 4. By a refined estimation, we determined that Q∗ = 3.75is the buckling critical value detected by the present method, which is very close to the critical value 3.7given in [Okada et al. 1979].

When the lateral pressure is small (Q ≤ Q∗= 3.75), the buckling does not occur; that is, there is neitherthe bifurcation point nor the unstable paths on the deflection-axial load curves. In fact, the deformationvaries continuously towards the deflection form of triple bulges with the large value of A3, with theincrease of the edge compression from the single bulge deflection form.

When the lateral pressure is large (Q > Q∗ = 3.75), the buckling occurs at a certain edge compression(the load at point A in Figure 10). In this case, the plate can show a jump behavior from a single bulgedeformation form to a three bulges deformation form once the edge pressure reaches the critical value. Inaddition, an external stimulus to the plate may switch the deflection form of the plate when edge pressureis in between load of point A and load of point B, because in this interval (hysteresis area) this systemhas two stable states either of which is physically realizable.

Figure 11 provides the solutions for plates 4 and 5 by the present method and the Rayleigh–Ritzmethod [Okada et al. 1979]. It can be seen that both methods can detect the critical buckling points.However, there exists a discrepancy between the two methods. Similar to the case in Section 4.3, thedifferences between the solutions are due to the different sets of boundary conditions, although both ofwhich might be loosely described as simply supported.

4.5. A rectangular plate subjected to shear and uniaxial compression. A rectangular plate under shearstress and uniaxial compression is analyzed with dimensions being a = 2, b = 1 and t = 0.009. Theuniaxial compression Px = 0.65Pxcr is exerted first and kept constant. The shear stress is applied in-crementally from 0.1τcr to 2τcr with a step size 0.1τcr. The final deformation of the plate is shown inFigure 12.


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

w/t

τ/τ

cr

Pure shear, 3× 3 present method



incremental Galerkin method (Ueda 1987)

Figure 13. Comparisons of load-deflection curves as given by present global Galerkinmethod and the incremental Galerkin method [Ueda et al. 1987] under a pure shearingload.

Load-deflection curves of the rectangular plate under pure shearing load are plotted in Figure 13 via thepresent method and the incremental Galerkin method. It is shown that the present method is in accordwith the incremental Galerkin method. The addition of terms in the assumed deflection function cangenerate a more accurate result if the result of the incremental Galerkin method is taken as a benchmark.Also, we can see that the deflection calculated by the present method is smaller than the incrementalGalerkin method in general. It can be seen that the buckling critical value for the plate is about τ ≈ τcr,since there is no extra force acting on the plate and the initial imperfection is very slight so that it doesnot influence much.

Load-deflection curves of the plate under combined uniaxial and shearing loadings are given inFigure 14. A similar conclusion can be obtained as the above pure shearing case. There is a discrepancybetween the results of the present method and the incremental Galerkin method. As the number ofterms in the deflection function increases, the difference between the present method and the incrementalmethod decreases. It shows from Figure 14 that the critical shearing value for the plate buckling decreasesfrom the theoretical value τcr of the pure shearing condition to approximately half of τcr of the combinedload condition. It illustrates that the uniaxial compression acting on the plate can degrade the criticalvalue for shearing buckling. Plus, with rectangular plates, more terms in the longitudinal direction arerequired such that the real deflection shape of the plate can be satisfactorily described.

4.5.1. Comparison of solvers. The consumed computing efforts via different solvers are provided inTable 6, where eight term (4× 2) deflection function is used in the present Galerkin method.

We see that the DNM2, that is, classical Newton method, is the fastest method both in terms ofiteration numbers and computational time. The DNM1, a variant Newton method, is comparable withthe Newton method. The DNMs are regarded as Newton-type methods, since they involve with the


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

w/t

τ/τ

cr

incremental Galerkin method (Ueda 1987)




Figure 14. Comparisons of load-deflection curves as given by present global Galerkinmethod and the incremental Galerkin method [Ueda et al. 1987] under uniaxial compres-sion combined with a shearing load.

method time(s) iterations TPI

MBECA1 174.07 17394 0.0100MBECA2 159.4 15938 0.0100DNM1 5.28 472 0.0112DNM2 5.05 457 0.0111DJIFM1 35.68 3479 0.0103DJIFM2 21.81 2143 0.0102

OVDA 12.96 1298 0.0100OIA/ODV[R] 95.03 9655 0.0098OIA/ODV[F] 16.19 1689 0.0096OIA(R) 12.93 1300 0.0099OIA(F) 12.94 1300 0.0100GOIA 12.94 1300 0.0100


inverse of Jacobian matrix. The TPIs of the DNM1 and DNM2 are larger than those of other methods,which can be explained by the need for inverting Jacobian matrix. In general, the iterative methods areseveral times faster than the continuous methods except the OIA/ODV[R]. However owing to the smallsize of the Jacobian matrix, the TPIs of DNMs are only slightly larger. Therefore the DNMs, which


converge quadratically fast, are preferable choices for solving the current Galerkin-resultant system ofNAEs which involve only a small number of nonlinear equations.

The prominent Jacobian-inverse-free methods, for example, OVDA, the OIAs and the GOIA, can beemployed in solving nonlinear problems whose initial guess is hard to choose, whose Jacobian matrixis ill-conditioned or nearly singular, when the number of equations tends to be very large, and when thesystem of equations is either over or under determined.

5. Conclusion

In this paper a global Galerkin method is presented for solving the simply supported initially imperfectvon Kármán plate under a combination of in-plane and lateral loads. The coupled nonlinear differentialequations are transformed into a system of NAEs via the global Galerkin method. The explicit formof the Jacobian matrix (“tangent stiffness” matrix) of the NAEs is derived to eliminate the symboliccalculations involved in solving the NAEs. Plates subjected to uniaxial compression, or lateral pressure,or a combination of axial compression and lateral pressure, or a combination of shear stress and axialcompression, are investigated by the present global Galerkin method in numerical examples. Largedeflections of plate under different loading conditions are analyzed, and nonlinear phenomena such asthe buckling behavior and the jump phenomenon are discussed. The present method is validated to be inagreement with the perturbation method, the incremental Galerkin method, the finite difference method,the finite element method and the experiments.

The present method is extremely advantageous provided that the deflection shape of a plate can beaccurately expressed by the deflection function with a few terms. Numerical experiments indicate thatthe computing effort of the present method would be very economic when less than forty terms areconsidered in the assumed deflection function. Also, because of the extremely high accuracy providedat a very modest cost, the global Galerkin method may also provide the much needed highly accuratebenchmark solutions against which other numerical methods may be validated. While distributed loadsare treated in the present paper using the Galerkin method and trigonometric basis functions, (nonlinear)plates subjected to concentrated loads at arbitrary locations will be treated in a forthcoming paper usingradial basis functions [Atluri 2005] and a spatial collocation or a Galerkin method. The resulting NAEsfor these point load problems can be solved by the same methods as in the present paper.

On the other hand, a series of scalar homotopy methods (mainly the Newton homotopy methods),which do not need to invert the Jacobian matrix, are reviewed and used to solve the Galerkin-resultingsystem of NAEs. The performance of each method is evaluated. The GOIA, the OIAs and the OVDApromise to be the best Jacobian-inverse-free methods hitherto. A general form equation is proposed toincorporate all the existing scalar Newton homotopy methods in a uniform framework. Interestingly, theclassical Newton method (labeled DNM2 in paper) can be generated from this general dynamical systemunder a certain condition.

Acknowledgement

This study is financially supported by the Doctorate Foundation of Northwestern Polytechnical Univer-sity (CX201305), the Chinese NSF (11172235), and the Doctor Subject Foundation of the Ministry ofEducation of China (20106102110018).


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Received 16 Oct 2013. Revised 8 Mar 2014. Accepted 7 Apr 2014.

HONGHUA DAI: [email protected] of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China

XIAOKUI YUE: [email protected] of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China

SATYA N. ATLURI: [email protected] for Aerospace Research and Education, University of California, Irvine, Irvine, CA 92697, United States

mathematical sciences publishers msp












http://dx.doi.org/10.1016/0045-7949(87)90067-8

http://dx.doi.org/10.1016/0045-7949(87)90067-8

http://ci.nii.ac.jp/naid/110003861539


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ADAIR R. AGUIAR University of São Paulo at São Carlos, BrazilKATIA BERTOLDI Harvard University, USA

DAVIDE BIGONI University of Trento, ItalyIWONA JASIUK University of Illinois at Urbana-Champaign, USA

THOMAS J. PENCE Michigan State University, USAYASUHIDE SHINDO Tohoku University, JapanDAVID STEIGMANN University of California at Berkeley

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J. P. CARTER University of Sydney, AustraliaR. M. CHRISTENSEN Stanford University, USAG. M. L. GLADWELL University of Waterloo, Canada

D. H. HODGES Georgia Institute of Technology, USAJ. HUTCHINSON Harvard University, USA

C. HWU National Cheng Kung University, TaiwanB. L. KARIHALOO University of Wales, UK

Y. Y. KIM Seoul National University, Republic of KoreaZ. MROZ Academy of Science, Poland

D. PAMPLONA Universidade Católica do Rio de Janeiro, BrazilM. B. RUBIN Technion, Haifa, Israel

A. N. SHUPIKOV Ukrainian Academy of Sciences, UkraineT. TARNAI University Budapest, Hungary

F. Y. M. WAN University of California, Irvine, USAP. WRIGGERS Universität Hannover, Germany

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Journal of Mechanics of Materials and StructuresVolume 9, No. 2 March 2014

Plane waves at the boundary of two micropolar thermoelastic solids with distinctconductive and thermodynamic temperatures

RAJNEESH KUMAR, MANDEEP KAUR and SATISH C. RAJVANSHI 121Dynamic compression of square tube cellular structures RYAN L. HOLLOMAN,

KARTHIKEYAN KANDAN, VIKRAM DESHPANDE and HAYDN N. G. WADLEY 149Dynamic response of twin lined shells due to incident seismic waves

J. P. DWIVEDI, V. P. SINGH and RADHA KRISHNA LAL 183Solutions of the von Kármán plate equations by a Galerkin method, without

inverting the tangent stiffness matrixHONGHUA DAI, XIAOKUI YUE and SATYA N. ATLURI 195

Bimaterial lattices with anisotropic thermal expansionMARINA M. TOROPOVA and CRAIG A. STEEVES 227

Origin and effect of nonlocality in a composite STEWART A. SILLING 245

JournalofMechanics

ofMaterials

andStructures

2014V

ol.9,No.2

http://dx.doi.org/10.2140/jomms.2014.9.121






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